The sum of two numbers is $100$, and their difference is $54$. What are the two numbers?
Answer: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 100}$ ${x-y = 54}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 154 $ $ x = \dfrac{154}{2} $ ${x = 77}$ Now that you know ${x = 77}$ , plug it back into $ {x+y = 100}$ to find $y$ ${(77)}{ + y = 100}$ ${y = 23}$ You can also plug ${x = 77}$ into $ {x-y = 54}$ and get the same answer for $y$ ${(77)}{ - y = 54}$ ${y = 23}$ Therefore, the larger number is $77$, and the smaller number is $23$.